Unified boundary conditions lead to supergroup WZW models.
problem Boundary conditions in topologically twisted Chern-Simons matter theories.
method Unified complexified gauge invariant boundary conditions.
result Low-energy effective field theories on M2-M5 intersections are supergroup WZW models.
The paper refines large N duality for knots using M-theory and string theory.
problem Understanding refined Chern-Simons invariants of knots.
method Formulating large N duality in refined Chern-Simons theory with torus knots, studying refined BPS states in M-theory, and relating to Type IIA string theory.
result The duality predicts graded dimensions of cohomology groups of moduli spaces of M2-M5 bound states associated to torus knots in the resolved conifold.
Generalizes intersection product for PL pseudomanifolds, proving duality.
problem Proving duality between intersection homology and cohomology products.
method Generalizes intersection product for PL pseudomanifolds, sheafifies the product, and corrects a proof of Poincaré duality.
result Establishes duality between intersection homology and cohomology products.
The paper examines how closed curves on surfaces intersect and how this intersection determines the curves.
problem Determining closed curves on surfaces based on their intersections.
method Constructing and studying k-equivalent curves, analyzing intersections with other curves. result Curves are determined by their intersections with all other curves, but non-simple curves require infinitely many intersections to distinguish.
3-manifold triangulation can be reconstructed from its intersection matrix.
problem Reconstructing the triangulation of 3-manifolds from their intersection matrix.
method Using the intersection matrix of a simplicial complex to determine the triangulation of a 3-manifold up to isomorphism.
result The intersection matrix is sufficient to determine the triangulation of a 3-manifold up to isomorphism.
Method calculates loop intersections on surfaces.
problem Computing intersections of loops on surfaces.
method Nielsen fixed point theory and Gröbner-Shirshov basis.
result Simple method to compute intersection numbers.
Introduces Lebesgue currents for real intersection theory.
problem Intersection of singular cycles in singular spaces.
method Introduces Lebesgue currents to define real intersection theory.
result Provides a foundation for real intersection theory.
Paper calculates properties of curves in 5D space.
problem Calculating geometric properties of curves in 5D space.
method Algorithms for differential geometric properties and geodesic curvature.
result Geometric properties of transversal intersection curves in R^5.
The paper finds diffeomorphic complex intersections with distinct Hodge numbers.
problem Identifying complex intersections with different Hodge numbers.
method Provided three pairs of 3-dimensional and one pair of 5-dimensional complex complete intersections, all diffeomorphic but with different Hodge numbers.
result Diffeomorphic complex intersections can have different Hodge numbers.
Study calculates intersection forms of almost-flat 4-manifolds.
problem Understanding the intersection forms of almost-flat 4-manifolds.
method Calculation of intersection forms for all 4-dimensional almost-flat manifolds.
result Intersection forms of all 4-dimensional almost-flat manifolds have been calculated.
New polynomials defined for virtual knots, calculated up to crossing 4.
problem Defining and calculating invariants for virtual knots.
method Intersection number of curves on a closed surface.
result Intersection polynomials calculated up to crossing 4.
Conditions for curves on a torus with specific pairwise intersections.
problem Finding curves on a torus with prescribed pairwise intersections.
method Necessary and sufficient conditions for curves on a torus with given pairwise intersections.
result Necessary and sufficient conditions for the existence of curves on a torus with specific pairwise intersections.
Study self-intersections of arcs on a pair of pants, proving natural number spectrum.
problem Understanding self-intersections of arcs on a pair of pants.
method Algorithm to compute self-intersection number, bounds established in terms of word length.
result Spectrum of self-intersection numbers covers all natural numbers.
Virtual knots with same writhe polynomial have equivalent intersection graphs.
problem Equivalence of intersection graphs for virtual knots.
method Proved equivalence through writhe polynomial.
result Intersection graphs of virtual knots with the same writhe polynomial are equivalent.
Study properties of self-similar continua with finite intersection property.
problem Characterize self-similar continua with finite intersection property.
method Prove intersection graph criterion, finite order theorem, and parameter matching theorem.
result All Jordan arcs starting from a intersection point in such continuum on a plane should have the same slope parameter at that point.
Paper predicts pedestrian trajectories at intersections with varying geometries.
problem Accurately predicting pedestrian trajectories at intersections with different geometries.
method Utilizes contravariant components of trajectories in curbside coordinate system and ASNSC formulation.
result Improves prediction accuracy by 7.2% in same train and test intersections.
Extended Birman-Series result to geodesics with infinitely many self-intersections.
problem Classifying geodesics with infinitely many self-intersections on hyperbolic surfaces.
method Defined a self-intersection function and extended the Birman-Series result to sets with bounded self-intersection functions.
result Same conclusion as Birman-Series for geodesics with self-intersection functions in o(l2). By considering a (not necessarily locally-flat) PL knot as the singular locus of a PL stratified pseudomanifold, we can use intersection homology theory to define intersection Alexander polynomials, a generalization of the classical Alexander polynomial invariants for smooth or PL locally-flat knots. We show that the i…
Estimates intersection pairing in hyperbolic 4-manifolds.
problem Estimating intersection pairing in hyperbolic 4-manifolds.
method Using Thurston norms of homology classes.
result Proved an estimate on intersection pairing.
James McClure recently showed that the domain for the intersection pairing of PL chains on a PL manifold M is a subcomplex of C∗(M)⊗C∗(M) that is quasi-isomorphic to C∗(M)⊗C∗(M) and, more generally, that the intersection pairing endows C∗(M) with the structure of a partially-defined commutati…
Classifies arcs on a 4-punctured sphere that intersect at most once.
problem Classifying arcs on a 4-punctured sphere with intersection constraints.
method Classification of maximal systems of arcs intersecting at most once.
result Maximal systems of arcs on the 4-punctured sphere identified.
Novel approach for large genus intersection number asymptotics.
problem Computing intersection numbers in large genus.
method Resurgent analysis of n-point functions with quantum curve.
result Extension of Aggarwal's results and new r-spin and Theta-class intersection numbers. The paper calculates self-intersections on a pair of pants using Bowen and Series' coding.
problem Computing the number of self-intersections of closed geodesics on a pair of pants.
method Used Bowen and Series' coding to compute self-intersections.
result Proved a conjecture and provided bounds for self-intersection numbers.
Generic potential primes have no self-intersections or intersections.
problem Finding non-degenerate periodic orbits without self-intersections.
method Generic convex Hamiltonian approach and Mañé genericity.
result Prime periodic orbits do not intersect or have self-intersections.
Study on shortest geodesics with self-intersections on hyperbolic surfaces.
problem Finding shortest closed geodesics with a fixed number of self-intersections.
method Investigated minimal length geodesics with at least k self-intersections, proving upper bounds and asymptotic behavior. result Self-intersection numbers are bounded by a linear function of k for any hyperbolic surface, and asymptotically like k in the presence of cusps. Diameters of ball intersections decrease as centers move apart.
problem Behavior of intersections of moving balls in Riemannian manifolds.
method Continuous decrease of intersection diameter as centers move apart.
result Diameter of intersections decreases continuously.
Paper proves curves can be smoothed to reduce self-intersection by exactly 1.
problem Prove that the shortest closed geodesic self-intersects exactly k times.
method Carefully smoothing intersection points reduces self-intersection by exactly 1.
result The shortest closed geodesic self-intersects exactly k times for hyperbolic and Riemannian metrics.
The paper establishes isomorphisms between different versions of intersection homology duality and products.
problem Comparing and relating different versions of intersection homology duality and products.
method Sheaf-theoretic and singular chain methods, PL pseudomanifolds, and de Rham isomorphism.
result Isomorphisms between sheaf-theoretic and singular intersection homology duality and products.
The paper extends intersection theory for b-divisors, proving monotonicity and volume inequalities.
problem Intersection theory for b-divisors and monotonicity of intersection products.
method Developed general intersection theory of nef b-divisors, defined restricted volume, proved monotonicity.
result Proved quantitative monotonicity of intersection product and new volume inequalities.
Study intersection polynomials of long virtual knots with supporting genera.
problem Characterize long virtual knots using geometric invariants.
method Define and analyze 1- and 2-supporting genera, and use them to filter long virtual knots. result Provide complete realizability criteria for all twelve intersection polynomials.
Paper computes Atiyah class for DG manifolds of amplitude +1.
problem Computing the Atiyah class for DG manifolds of specific amplitude.
method Computed the Atiyah class by encoding the derived intersection of sections and zero sections of vector bundles.
result Atiyah class vanishes if and only if the intersection is clean.
We continue here the investigation of the relationship between the intersection of a pair of subgroups of a Kleinian group, and in particular the limit set of that intersection, and the intersection of the limit sets of the subgroups. Of specific interest is the extent to which the intersection of the limit sets being …
Minimum algebraic intersection found in hyperbolic surfaces, growing with genus.
problem Finding the minimum algebraic intersection form in hyperbolic surfaces.
method Analyzing algebraic intersection form in moduli space of hyperbolic surfaces.
result Minimum grows in the order of (logg)−2 with genus. Paper explores topological invariance in torsion-sensitive intersection homology.
problem Torsion-sensitive intersection homology's topological invariance.
method Analogous to classical intersection homology, investigates invariance of sheaf complexes.
result Provides torsion-sensitive versions of existing invariance theorems and new results.
In this paper we study relations between intersection numbers on moduli spaces of curves and Hurwitz numbers. First, we prove two formulas expressing Hurwitz numbers of (generalized) polynomials via intersections on moduli spaces of curves. Then we show, how intersection numbers can be expressed via Hurwitz numbers. An…
New method bounds intersections of exact Lagrangians in cotangent bundles.
problem Counting intersections of exact Lagrangian submanifolds in cotangent bundles.
method Sheaf quantization in Tamarkin's category for clean and degenerate intersections.
result Cardinality of intersections is bounded by sheaf Hom spaces.
The paper tackles intersectional fairness in machine learning, proposing methods to assess and mitigate bias.
problem Achieving optimal predictive performance is not enough; fairness with respect to multiple sensitive attributes is crucial.
method The paper introduces a comprehensive framework for auditing and achieving intersectional fairness in classification problems, including metrics, estimation methods, and post-processing techniques.
result The proposed methods can robustly estimate and mitigate intersectional bias in classification models without compromising predictive performance.
Study the intersection of positive closed currents using tangent currents and King's residue formula.
problem Investigate the intersection of positive closed currents in complex manifolds.
method Employ tangent currents and King's residue formula to establish a natural condition for intersection.
result Derive an integral representation of the intersection of positive closed currents.
New method for decomposing surfaces with minimal intersecting filling pairs.
problem Decomposing surfaces with minimal intersecting filling pairs.
method Explicit algebraic means to determine homeomorphism class and decomposition criteria.
result Explicit means to determine homeomorphism class and decomposition criteria for surfaces with minimally intersecting filling pairs.
Unified quantum invariants via intersections of embedded Lagrangians.
problem Unified quantum invariants for Uq(sl(2)). method State sum of Lagrangian intersections in configuration spaces.
result Recovery of coloured Jones and Alexander polynomials.
Positivity of intersections in 4-manifolds leads to taming symplectic structures.
problem Taming symplectic structures in almost complex 4-manifolds.
method Proof of positivity of intersections of pseudoholomorphic curves.
result Positivity of intersections is stable and leads to taming symplectic structures.
The paper explores intersectional fairness in machine learning, proving bounds on it.
problem Intersectional fairness in machine learning, especially when multiple protected attributes are involved.
method Statistical analysis and bounds on intersectional fairness, leveraging marginal fairness.
result Theoretical bounds on intersectional fairness can be computed from marginal fairness and other statistical quantities.
Proves a conjecture about Lagrangian intersections using new theory.
problem Homological Arnol'd conjecture on Lagrangian intersections.
method New Lagrangian Ljusternik-Schnirelman theory and fundamental quantum factorizations.
result Uniform lower bounds on Lagrangian intersection numbers.
Max arcs on spheres intersecting twice, proven.
problem Maximizing arcs on spheres with intersection constraints.
method Proved maximal cardinality of arcs intersecting at most twice.
result Maximal cardinality of arcs intersecting at most twice is |X|(|X| + 1)(|X| + 2).
The paper proves the exact number of singular points in the intersection of convex shapes.
problem Determining the exact number of singular points in the intersection of convex shapes.
method Analyzing the intersections of n translates of a strictly convex, smooth, convex body in the Euclidean plane.
result The intersection of n translates of a convex body has exactly n points of singularity along its boundary.
Proves intersection properties of minimal hypersurfaces in various spaces.
problem Intersection properties of minimal hypersurfaces in different geometric settings.
method Two approaches: classifications of stable minimal hypersurfaces and conformal change with comparison geometry.
result Intersection properties for minimal hypersurfaces in specific geometric settings, including free boundary minimal hypersurfaces.
The paper extends the avoidance principle for mean curvature flows, proving new intersection dimension monotonicity results.
problem Understanding the behavior of intersections in mean curvature flows.
method Proving new intersection dimension monotonicity results for mean curvature flows, Brakke flows, and level set flows.
result The dimension of the intersection of mean curvature flows is non-increasing over time.
Study joins and intersections of subgroups in free groups, disproving and repairing a lemma.
problem Disproving and repairing a lemma on joins and intersections of subgroups in free groups.
method Analyzing graphs of joins and intersections of finitely generated subgroups of a free group.
result Showed how to disprove and repair a lemma of Imrich and Müller on these graphs.